A propos des matrices aléatoires et des fonctions L

The introduction is a survey about the origins and limits of analogies between random matrices in the compact groups and L-functions. We then state the main results of this thesis.

The first two chapters give a probabilistic flavor of results by Keating and Snaith, previously obtained by analytic methods. In particular, a common framework is set in which the notion of independence naturally appears from the Haar measure on a compact group. For instance, if $g$ is a random matrix from a compact group endowed with its Haar measure, $\det(\Id-g)$ may be decomposed

as a product of independent random variables.

Such independence results hold for the Hua-Pickrell measures, which generalize the Haar measure. Chapter 3 focuses on the point process induced on the spectrum

by these laws on the unit circle : these processes are determinantal with an explicit kernel, called the hypergeometric kernel. The universality of this kernel is

then shown : it appears for any measure with asymmetric singularities.

The characteristic polynomial of random matrices can be considered as an orthogonal polynomial associated to a spectral measure. This point of view combined with the widely developed theory of orthogonal polynomials on the unit circle yields results about the (asymptotic) independence of characteristic polynomials,

a large deviations principle for the spectral measure and limit theorems for derivatives and traces. This is developed in Chapters 4 and 5.

Chapter 6 concentrates on a number theoretic issue: it contains a central limit theorem for $\log \zeta$ evaluated at distinct close points. This implies correlations when counting the zeros of $\zeta$ in distinct intervals at a mesoscopic level, confirming numerical experiments by Coram and Diaconis. A similar result holds for random matrices from the unitary group, giving further insight

for the analogy at a local scale.

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Contributor : Paul Bourgade < decryptMail("cnhyobhetnqr@tznvy.pbz", "link5d82a97638e32", "") >

Submitted on : Tuesday, April 7, 2009 - 10:58:11 AM

Last modification on : Friday, January 4, 2019 - 5:32:33 PM

Long-term archiving on : Thursday, June 10, 2010 - 7:56:20 PM

Contributor : Paul Bourgade <

Submitted on : Tuesday, April 7, 2009 - 10:58:11 AM

Last modification on : Friday, January 4, 2019 - 5:32:33 PM

Long-term archiving on : Thursday, June 10, 2010 - 7:56:20 PM